# Antidifferentiation

Antidifferentiation is the process of, given a function $$f(x)$$, finding another function $$F(x)$$ whose derivative is the given function, $$\frac{d}{dx} F(x) = F^\prime(x) = f(x)$$. Antidifferentiation and the antiderivative are part of a larger process called integration. Formally, the antiderivative of $$f(x)$$ is the set of functions $$F(x) + C$$ such that $\frac{d}{dx}[F(x) + C] = f(x)$ where $$C$$ is what’s called the constant of integration. This can be restated as follows: if two functions $$F(x)$$ and $$G(x)$$ have the same derivative $$f(x)$$, then $$F(x)$$ and $$G(x)$$ differ by at most a constant $$F(x) = G(x) + C$$. …

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# Economics Applications of Derivatives

Some interesting things you can do with derivatives in the context of economics are to: Find the price elasticity of a demand function. Find the maximum of a total-revenue function. Characterize demand in terms of elasticity. Retailers and manufacturers often need to know how a small change in price will affect the demand of a product. If a small increase in price produces no change in demand, then the price increase is a reasonable decision to make. …

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# Derivatives of Exponential and Logarithmic Functions

The derivative of an exponential function of the form $$a^x$$ is $\frac{d}{dx}a^x = (\ln a)a^x$ from sympy import * from sympy.abc import x expr = 2**x expr2 = 3**(2*x) diff(expr) ## 2**x*log(2) diff(expr2) ## 2*3**(2*x)*log(3) The derivative of a function of the form $$\log_ax$$is $\frac{d}{dx}\log_a x = \frac{1}{\ln a} \cdot \frac{1}{x}$ expr3 = log(x, 8) expr4 = log((x**2 + 1), 3) diff(expr3) ## 1/(x*log(8)) diff(expr4) ## 2*x/((x**2 + 1)*log(3))

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# Exponential Decay

In the equation of population growth, $$dP/dt = kP$$, the constant $$k$$ is given by $$k = (\text{birth rate}) - (\text{death rate})$$. Thus, a population grows only when the birth rate is greater than the death rate. When the death rate is greater than the birth rate we have a decrease or “decay”. Our equation for population decay becomes $$dP/dt = -kP$$, where $$k >0$$. The function that satisfies that rate of return is then …

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We go over some textbook exercises for exponential functions. Franchise Expansion A franchise business is selling franchises for pizza shops throughout the country. The marketing manager estimates that the number of franchises, $$N$$, will increase at the rate of 10% per year, that is, $\frac{dN}{dt} = 0.10N$ Find the function that satisfies this equation, assuming that the number of franchises at $$t = 0$$ is 50. How many franchises will be there in 20 years? …